What is the moment of inertia? What is the moment of inertia measured? Task example

Mechanics is one of the main branches of physics, in which the laws of motion and equilibrium of bodies are considered. In the quantitative description of the motion of rotation, the moment of inertia is an important quantity. In this article, we study this value. In addition, we answer the question of what measures the moment of inertia of a solid.

The concept of the moment of inertia for the point of material

As a physical quantity, it is determined as the product of mass and the square of the radius of rotation. Suppose that there is some material point that has mass m. It rotates around the axis, while the radius of the circle is r. Under given inertia conditions, the moment is calculated in accordance with the following formula:

I = m * r ² .

This formula can be used even in cases of studying bodies with a complex shape. The main condition for the equality to be true is the presence of a huge difference between the distance to the axis of rotation r and the geometric dimensions of the body itself. For example, when calculating the value of I for our planet, which rotates around the Sun in a circular path, we can consider the Earth as a material point, since the distance to the star is several orders of magnitude greater than the radius of the planet.

The value of I for a body of arbitrary shape

If the geometric dimensions of the rotating body are slightly different from the radius r, then the shape of the body should be taken into account. Given the named factor, the moment of inertia is calculated using the following formula:

I = ∫ _m (r ² * dm).

In fact, this equality is the sum of the moments of inertia of all the material points that form the body. When conducting practical calculations, the written formula is used in a slightly different form, which is presented below:

I = ∫ _V (ρ * r ² * dV).

As you can see, integration over mass m is replaced by integration over volume V. Here, the Greek letter ρ denotes density. If the body is homogeneous, then ρ will be a constant value that can be taken out of the sign of the integral. If the mass is nonuniformly distributed over the body, then the density will be a function of the parameter r. The recorded formula is convenient to use when determining I of different bodies, because the calculation is performed using the mental division of the body into elementary volumes dV.

The results of applying the above equality for geometric bodies of perfect shape, for example, for a sphere, cylinder or rod, are collected in the corresponding tables. What is the moment of inertia measured? The figure below shows the values ​​of I for some bodies. As you can see, all the formulas linearly depend on the mass of bodies and on the square of the geometric parameter.

What is the measured moment of inertia of the body?

Having received the necessary theoretical information for the quantity I, everyone can easily answer the question posed. Indeed, if you look at the formula for the I material point, then, answering the question of what is measured by the moment of axial inertia, it should be answered in kilograms per square meter. Abbreviated as this unit is written kg * m ² . Obviously, we will get the same unit if we use the integral expression in terms of volume and density.

Note that kg * m ² can also be written as m ² * kg. This form of recording is also allowed, however, it is not used in practical physics.

Since both a kilogram and a meter are system units for measuring mass and length, respectively, kg * m ² is also a SI unit for the moment of inertia.

The study unit should not be confused with another, which is designated as kg / m ² . Although it is rarely used, and it is not a SI unit, nevertheless, it allows one to calculate the corresponding pressure if it is multiplied by the acceleration of gravity.

Task example

Having figured out what the moment of inertia is measured and how to calculate it, we will solve the following problem: it is necessary to determine the moment of inertia of the Earth, assuming it as a material point.

To successfully solve this problem, you should know only two parameters: the mass of the planet and the average radius of its solar orbit. Both values ​​can be found in the respective directories. The mass M and the radius of the Earth’s orbit R are equal to:

M = 5.972 * 10 ²⁴ kg;

R = 149.6 * 10 ⁹ m.

Using the expression to calculate the inertia of the moment of the point material, we arrive at the following result:

I = M * R ² = 5.972 * 10 ²⁴ * (149.6 * 10 ⁹ ) ² = 1.34 * 10 ⁴⁷ kg * m ² .

We got gigantic value. If we compare it with the moment of inertia of the Earth relative to its own axis, it turns out that it will be a billion times smaller than the calculated value. Thus, the approximation of a material point is quite appropriate for the considered problem.